TransientSchr¨odinger-Poisson Simulations of a High-Frequency Resonant Tunneling Diode Oscillator Jan-Frederik Mennemann ∗ Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8–10, 1040 Wien, Austria Ansgar J¨ ungel Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8–10, 1040 Wien, Austria Hans Kosina Institute for Microelectronics, Vienna University of Technology, Gusshausstraße 27–29, 1040 Wien, Austria Abstract Transient simulations of a resonant tunneling diode oscillator are presented. The semi- conductor model for the diode consists of a set of time-dependent Schr¨ odinger equations coupled to the Poisson equation for the electric potential. The one-dimensional Schr¨ odinger equations are discretized by the finite-difference Crank-Nicolson scheme using memory-type transparent boundary conditions which model the injection of electrons from the reservoirs. This scheme is unconditionally stable and reflection-free at the boundary. An efficient re- cursive algorithm due to Arnold, Ehrhardt, and Sofronov is used to implement the trans- parent boundary conditions, enabling simulations which involve a very large number of time steps. Special care has been taken to provide a discretization of the boundary data which is completely compatible with the underlying finite-difference scheme. The tran- sient regime between two stationary states and the self-oscillatory behavior of an oscillator circuit, containing a resonant tunneling diode, is simulated for the first time. Keywords: Schr¨ odinger-Poisson system, transient simulations, discrete transparent boundary conditions, resonant tunneling diode, high-frequency oscillator circuit 2000 MSC: 65M06, 35Q40, 82D37 * Corresponding author Email addresses: [email protected](Jan-Frederik Mennemann), [email protected](Ansgar J¨ ungel), [email protected](Hans Kosina) Preprint submitted to Journal of Computational Physics October 20, 2012
Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße
8–10, 1040 Wien, Austria
Ansgar Jungel
Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße
8–10, 1040 Wien, Austria
Hans Kosina
Institute for Microelectronics, Vienna University of Technology, Gusshausstraße 27–29, 1040 Wien,
Austria
Abstract
Transient simulations of a resonant tunneling diode oscillator are presented. The semi-conductor model for the diode consists of a set of time-dependent Schrodinger equationscoupled to the Poisson equation for the electric potential. The one-dimensional Schrodingerequations are discretized by the finite-difference Crank-Nicolson scheme using memory-typetransparent boundary conditions which model the injection of electrons from the reservoirs.This scheme is unconditionally stable and reflection-free at the boundary. An efficient re-cursive algorithm due to Arnold, Ehrhardt, and Sofronov is used to implement the trans-parent boundary conditions, enabling simulations which involve a very large number oftime steps. Special care has been taken to provide a discretization of the boundary datawhich is completely compatible with the underlying finite-difference scheme. The tran-sient regime between two stationary states and the self-oscillatory behavior of an oscillatorcircuit, containing a resonant tunneling diode, is simulated for the first time.
Preprint submitted to Journal of Computational Physics October 20, 2012
1. Introduction
The resonant tunneling diode has a wide variety of applications as a high-frequency andlow-consumption oscillator or switch. The resonant tunneling structure is usually treated asan open quantum system with two large reservoirs and an active region containing a double-barrier heterostructure. Accurate time-dependent simulations are of great importanceto develop efficient and reliable quantum devices and to reduce their development timeand cost. There exist several approaches in the literature to model a resonant tunnelingdiode. The simplest approach is to replace the diode by an equivalent circuit containingnonlinear current-voltage characteristics [28]. Another approximation is to employ theWannier envelope function development [29]. Other physics-based approaches rely on theWigner equation [12, 27], the nonequilibrium Green’s function theory [15, 26], quantumhydrodynamic models [19, 21, 24], and the Schrodinger equation [11, 13, 31, 32].
In this paper, we adopt the latter approach and simulate the time-dependent behaviorof a resonant tunneling diode using the Schrodinger-Poisson system in one space dimension.In this setting, the electrons are assumed to be in a mixed state with Fermi-Dirac statisticsand the electrostatic interaction is taken into account at the Hartree level. Each stateis determined as the solution of the transient Schrodinger equation with nonhomogeneoustransparent boundary conditions. The Schrodinger equations are discretized by the Crank-Nicolson finite-difference scheme and coupled self-consistently to the Poisson equation. Themain originality of this paper is the adaption (and slight improvement) of existing numericaltechniques from, e.g., [1, 6, 31], to a long-time simulation of a high-frequency oscillatorcircuit containing a resonant tunneling diode. Our changes in the techniques are necessaryto achieve simulations without spurious oscillations in the numerical transient solution. Inthe following, we detail the techniques used as well as the novel features.
First, we consider the one-dimensional stationary problem, since it builds the basis forthe transient simulations. The stationary transparent boundary conditions are discretizedin such a way that their discrete version is compatible with the underlying finite differencediscretization, as proposed in [4]. Thereby, any (numerical) spurious oscillation is elim-inated, which would otherwise propagate in the transient simulations. In the literature[10, 31], a modified version of the potential energy is used to overcome problems of nu-merical convergence. Physically interpreted, this model introduces artificial surface chargedensities at the junction interfaces of the tunneling diode. We are able to solve the originalproblem. This represents an improvement compared to the simulations in [31], where themodified model is also employed for the time-dependent case.
Second, the time-dependent Schrodinger-Poisson system with transparent boundaryconditions is approximated. Since these boundary conditions are of memory type [4, 8],their numerical implementation requires to store (and to use) the boundary data for allthe past history. For this reason, simulations involving longer time scales are extremelycostly. This explains why simulations in the literature [10, 13, 31] have been restricted tosome picoseconds only. We solve this problem by using a fast evaluation of the discreteconvolution kernel of sum-of-exponentials, which has been presented in [6] and employedin [5] on circular domains. To our knowledge, this rather new numerical technique has not
2
been applied to realistic device simulations so far.A challenge in the transient simulations results from the large number of wave func-
tions which need to be propagated, accounting for the energy distribution of the incomingelectrons. Each state is provided with transparent boundary conditions, which raises thecomputational cost sharply. To cope with a large number of Schrodinger equations tobe solved, we developed a parallel version of our solver utilizing multiple cores on sharedmemory processors. This enables us to present, for the first time, simulations to theSchrodinger-Poisson system for large times up to 100 ps (ps = picosecond) with reasonablecomputational effort (compared to 5 ps in [31], 6 ps in [13], and 8 ps in [10]).
Another novelty in this paper concerns the discretization of nonhomogeneous discretetransparent boundary conditions. They are necessary to describe continuously varyingapplied potentials (as in an oscillator circuit). It is well known that, using a suitablegauge change, one can get rid of the transient potential [2]. Corresponding nonhomoge-neous transparent boundary conditions can be found in [10]. In numerical simulations,however, we observed that these boundary conditions may lead to unphysical distortionsin the conduction current density. The reason is that the considered discretization of thegauge change is not compatible with the underlying finite-difference scheme. Therefore,we suggest a new discretization which is derived from the Crank-Nicolson time integra-tion scheme. Our approach completely removes these numerical artifacts, and we showthat the total current density is now perfectly conserved. We stress the fact that our dis-cretization is completely consistent with the underlying Crank-Nicolson scheme inheritingits conservation and stability properties.
Third, the numerical results allow us to identify plasma oscillations in a certain timeregime of the resonant tunneling diode and to estimate the life time of the resonant state.We present, for the first time, simulations of a high-frequency oscillator circuit containinga reconant tunnelding diode, based on a full Schrodinger-Poisson solver with transparentboundary conditions. Simplified tunneling diode oscillators have been considered in [28,29, 30]. Our approach enables us to observe the complex spatio-temporal behavior ofmacroscopic quantities inside the resonant tunneling diode in an unprecedented way.
The paper is organized as follows. In Section 2, we detail the algorithm of the station-ary problem. The transient algorithm is described in Section 3. In Section 4 we considernumerical experiments for constant applied voltage and time-dependent applied voltage.Furthermore, the numerical convergence related to the approximation of the discrete con-volution kernel by sum-of-exponentials is investigated. Finally, we present high-frequencyoscillator circuit simulations in Section 5.
2. Stationary simulations
The steady state is the basis for the transient simulations. Therefore, we discuss firstthe stationary regime.
3
2.1. Schrodinger-Poisson model
We assume that the one-dimensional device in (0, L) is connected to the semi-infiniteleads (−∞, 0] and [L,∞). The leads are assumed to be in thermal equilibrium and atconstant potential. At the contacts, electrons are injected with some given profile. Wesuppose that the charge transport is ballistic and that the electron wave functions evolveindependently from each other. The one-dimensional device consists of three regions: twohighly doped regions, [0, a1] and [a6, L], with the doping concentration n1
D and a lowlydoped region, [a1, a6], with the doping density n2
D (see Figure 1). The middle intervalcontains a double barrier, described by the barrier potential
Figure 1: Barrier potential and doping profile of a double-barrier heterostructure.
The Coulomb interaction is taken into account at the Hartree level, i.e. by an infinitenumber of Schrodinger equations
− ~2
2m
d2φk
dx2(x) + V (x)φk(x) = E(k)φk(x), x ∈ R, (1)
4
self-consistently coupled to the Poisson equation,
−d2Vselfdx2
=e2
ε(n[Vself ]− nD), x ∈ (0, L),
Vself(0) = 0, Vself(L) = −eU,(2)
where V = Vbarr + Vself is the potential energy. The physical parameters are the reducedPlanck constant ~, the effective electron mass m, the elementary charge e, and the permit-tivity ε = εrε0, being the product of the relative permittivity εr and the electric constantε0. Furthermore, U ≥ 0 denotes the applied voltage, and the electron density is defined by
n[Vself ](x) =
∫
R
g(k)|φk(x)|2dk. (3)
The injection profile g(k) is given according to Fermi-Dirac statistics by
g(k) =mkBT02π2~2
ln
(1 + exp
(EF − ~
2k2/(2m)
kBT0
)), (4)
where kB is the Boltzmann constant, T0 the temperature of the semiconductor and EF theFermi energy (relative to the conduction band edge). In all subsequent simulations, weuse, as in [31], εr = 11.44, T0 = 300K, EF = 6.7097 · 10−21 J, and the effective mass ofGallium arsenide, m = 0.067me, with me being the electron mass at rest.
In order to define the total electron energy E(k) depending on the wave number k ∈ R,we need to distinguish the cases k > 0 and k < 0. For k > 0, the electrons enter from theleft, and we have E(k) = ~
2k2/(2m). The wave function in the leads is given by
φk(x) = eikx + r(k)e−ikx, x < 0,
φk(x) = t(k) exp(i√
2m(E(k)− V (L))/~2x), x > L.
Eliminating the transmission and reflection coefficients t(k) and r(k), respectively, theboundary conditions
φ′
k(0) + ikφk(0) = 2ik, φ′
k(L) = i√
2m(E(k)− V (L))/~2φk(L) (5)
follow. For k < 0, the electrons enter from the right. The total energy is given byE(k) = ~
2k2/(2m)− eU , and the wave function in the leads reads as
φk(x) = t(k) exp(−i√
2mE(k)/~2x), x < 0,
φk(x) = eikx + r(k)e−ikx, x > L.
This yields the boundary conditions
φ′
k(0) = −i√
2mE(k)/~2φk(0), φ′
k(L) + ikφk(L) = 2ikeikL. (6)
Summarizing, the stationary problem consists in the Schrodinger equation (1) withthe transparent boundary conditions (5)-(6) coupled to the Poisson equation (2) via theelectron density (3). We remark that the existence and uniqueness of solutions to aSchrodinger-Poisson boundary-value problem similar to (1)-(6) has been shown in [9].
5
2.2. Discrete transparent boundary conditions
We recall the finite-difference discretization of the stationary Schrodinger equation withtransparent boundary conditions [4]. Using standard second-order finite differences on theequidistant grid xj = jx, j ∈ 0, . . . , J, with xJ = L and x > 0, we find for the gridpoints located in the computational domain,
φj+1 − 2φj + φj−1 +2m(x)2
~2(E(k)− Vj)φj = 0. (7)
It is well known that a standard centered finite-difference discretization of the bound-ary conditions (5) and (6) may lead to spurious oscillations in the numerical solution [4].In principle, the numerical errors can be made as small as desired by choosing x suffi-ciently small. However, since the stationary solutions will serve as intitial states in ourtransient simulations, we need to avoid any spurious oscillation, which would otherwise bepropagated with every time step.
For this, we apply (stationary) discrete transparent boundary conditions compatiblewith the finite-difference discretization (7) as proposed in [4]. For the sake of complete-ness, we review the derivation. Note that the final discretization is equivalent to thediscretization (7) extended to the whole space, i.e. for j ∈ Z.
In the semi-infinite leads j ≤ 0 and j ≥ J , the potential energy is assumed to beconstant,
Vj =
V0 = 0 for j ≤ 0,
VJ = −eU for j ≥ J.
Then (7) reduces to a difference equation with constant coefficients which admits twosolutions of the form φj = (α±
0,J)j , where
α±
0,J = 1− m(E(k)− V0,J)(x)2~2
± i√
2m(E(k)− V0,J)(x)2~2
− m2(E(k)− V0,J)2(x)4~4
.
Here, E(k) − V0,J corresponds to the kinetic energy Ekin0,J (k) in the left or right lead. In
case Ekin0,J (k) > 0, the solution is a discrete plane wave and (x)2 < 2~2/(m(E(k)− V0,J))
is needed to ensure |α0,J | = 1, which in practise is not a restriction. In case Ekin0,J (k) = 0,
the solution is constant. Depending on the applied voltage, Ekin0,J (k) might also become
negative. In that case, the solution is decaying or growing exponentially fast and we selectthe decaying solution as it is the only physically reasonable solution.
In practice, we start with the calculation of the total energy E(k) = Ekin0,J (k)+V0,J . For
electrons coming from the left contact we have E(k) = Ekin0 (k). As the incoming electron
is represented by a discrete plane wave, Ekin0 (k) is positive but, depending on the applied
voltage, EkinJ (k) might be positive, zero or negative. For electrons coming from the right
contact, we have E(k) = EkinJ (k) − eU . Again, the incoming wave function is a discrete
plane wave, i.e., EkinJ (k) > 0 but nothing is said about Ekin
0 (k). At this point it should be
6
noted that the kinetic energy of the incoming electron needs to be computed according tothe discrete dispersion relation
Ekin(k) =~2
m(x)2 (1− cos(kx)),
which follows after solving the centered finite-difference discretization of the free Schrodin-ger equation
− ~2
2m
d2
dx2eikx = Ekin(k)eikx.
In the limit x→ 0, we recover the continuous dispersion relation Ekin(k) = ~2k2/(2m).
Let us consider a wave function entering the device from the left contact (k > 0). Forj ≤ 0, the solution to (7) is a superposition of an incoming and a reflected discrete planewave, φj = βj +Bβ−j, where β = α0. We eliminate B from φ−1 = β−1 +Bβ, φ0 = 1 +Bto find the discrete transparent boundary condition at x0:
−β−1φ−1 + φ0 = 1− β−2.
For j ≥ L, the solution to (7) is given by φj = Cγj with γ = αJ . This means thatφJ+1 = CγJ+1 = γφJ , and the boundary condition at xJ becomes
φJ − γ−1φJ+1 = 0.
Summarizing, we obtain the linear system Aφ = b with the tridiagonal matrix A consist-ing of the main diagonal (−β−1,−2+ 2m(x)2(E(k)−V0)/~2, . . . ,−2+ 2m(x)2(E(k)−VJ)/~
2,−γ−1) and the first off diagonals (1, . . . , 1). The vector of the unknowns is givenby φ = (φ−1, . . . , φJ+1)
⊤ and b represents the right-hand side b = (1− β−2, 0, . . . , 0)⊤.The case of a wave function entering from the right contact (k < 0) works analogously.
2.3. Solution of the Schrodinger-Poisson system
We explain our strategy to solve the coupled Schrodinger-Poisson system. To this end,we introduce the equidistant energy grid
K = −kM ,−kM +k, . . . ,−k,+k, . . . , kM −k, kM, K := |K|. (8)
The electron density (3) is approximated by
ndisc[Vself ](x) = k∑
k∈K
g(k)|φk(x)|2,
where the Fermi-Dirac statistics g(k) is defined in (4) and the functions φk are the scat-tering states, i.e., the solutions to the discretized stationary Schrodinger equation (7) withdiscrete transparent boundary conditions as described in Section 2.2. This approximationis reasonable if k is sufficiently small and kM is sufficiently large. In the numerical simu-lations below, we choose K = 3000 and, as in [10, Section 5], kM =
√2m(EF + 7kBT0)/~,
recalling that EF = 6.7097 · 10−21 J and T0 = 300K.
7
The discrete Schrodinger-Poisson system is iteratively solved as follows. We set V =Vbarr+ V
(p)self,U , where V
(p)self,U is the p-th iteration of Vself for the applied voltage U . Given V ,
we compute a set of quasi eigenstates φ(p)k k∈K. This defines the discrete electron density
ndisc[V(p)self,U ] = k
∑
k∈K
g(k)|φ(p)k (x)|2.
The Poisson equation is solved by employing a Gummel-type method [20]:
− d2
dx2V
(p+1)self,U =
e2
ε
(n[V
(p)self,U ] exp
(V
(p)self,U − V
(p+1)self,U
V refself
)− nD
),
V(p+1)self,U (0) = 0, V
(p+1)self,U (L) = −eU.
The idea of the Gummel method is to decouple the Schrodinger and Poisson equationsbut to formulate the Poisson equation in a nonlinear way, using the relation betweenthe electron density and electric potential in thermal equilibrium. The parameter V ref
self
can be tuned to reduce the number of iterations; we found empirically that the choiceV refself = 0.04 eV minimizes the iteration number. If the relative error in the ℓ2-norm is
smaller than a fixed tolerance,
∥∥∥∥∥V
(p+1)self,U − V
(p)self,U
V(p+1)self,U
∥∥∥∥∥2
≤ δ, (9)
we accept Vself := V(p+1)self,U and φ(p+1)
k k∈K as the approximate self-consistent solution. Oth-erwise, we proceed with the iteration p + 1 → p + 2 and compute a new set of scatteringstates. The procedure is repeated until (9) is fulfilled. We have choosen the toleranceδ = 10−6.
For zero applied voltage we use V(0)self,0mV = 0mV to start the iteration. Only 7 iterations
are needed until criterion (9) is fulfilled. As a result we obtain V(7)self,0mV, which is depicted
in Figure 2(a) (solid line).Numerical problems arise when non-equilibrium solutions are computed. As an example
we consider the case of a small applied voltage U = 1mV. To start the iteration process weuse the previously computed solution, i.e., we set V
(0)self,1mV := V
(7)self,0mV. The next iterations
are illustrated in Figure 2(a) (dashed lines). Obviously they do not converge and arephysically not realistic. This phenomenon is a well-known in the literature [17, 18] andis believed to be related to the absence of inelastic processes in the Schrodinger-Poissonequations.
In the literature [10, 31], a modified version of the Schrodinger-Poisson equations isemployed to overcome this problem. The modification concerns the description of thepotential energy in the Poisson equation. For this, we write the Poisson equation (2) as
8
−200
0
200
400
potentialen
ergyin
meV
0 50 85 135position in nm
−50
0
50
100
(a)
−400
−200
0
200
400
potentialen
ergyin
meV
0 50 85 135position in nm
−100
−50
0
50
100
(b)
Figure 2: (a) Solid line: self-consistent solution Vself for U = 0mV, found after 7 iterations.Dashed lines: divergent approximations for U = 1mV. Dotted line: barrier potential. (b)Solid line: self-consistent solution V1 for U = 250mV according to approximation (10).Dotted line: sum of the barrier potential and a ramp-like potential.
follows:
− d2V0dx2
= 0 in (0, L), V0(0) = 0, V0(L) = −eU,
− d2V1dx2
=e2
ε(n− nD) in (0, L), V1(0) = 0, V1(L) = 0,
i.e., the self-consistent potential is Vself = V0+V1. The first boundary-value problem can besolved explicitly: V0(x) = −eUx/L, x ∈ [0, L]. In [10, 31], the linearly decreasing potentialV0 has been replaced by the ramp-like potential
V0(x) = −B0
(x− a1a6 − a1
1[a1,a6) + 1[a6,∞)
), x ∈ [0, L], (10)
where 1I is the characteristic function on the interval I ⊂ R (see Figure 1 for the definition
of a1 and a6). The function V0 + Vbarr is illustrated in Figure 2(b) (dotted line). The
potential energy is then given by V = V0 + V1 + Vbarr. Using this modified physical model,the above Gummel iteration scheme for the Poisson equation for V1 converges without anyproblems, see Figure 2(b) (solid line), even for large applied voltages. However, we willsee below that the results from the modified model differ considerably from the resultsobtained by the original Schrodinger-Poisson model. Furthermore, the potential energy isno longer differentiable at a1 and a6. This may be interpreted as a model of surface chargedensities at the interfaces which, however, are not intended in the model.
In fact, we are able to solve numerically the original Schrodinger-Poisson problem. Tothis end, the applied voltage needs to be increased in small steps. We found that thestarting potential in each step needs to be initialized carefully. More precisely, given the
9
self-consistent solution Vself,U for the applied voltage U , we wish to compute a self-consistentsolution with the applied voltage U +U . In each step we choose
V(0)self,U+U(x) := Vself,U(x)−U
2x− LL
1[L/2,L] (11)
to start the iteration. For U = 0mV and U = 25mV, the Gummel scheme convergesto a physically reasonable solution after 7 iterations (i.e., (9) is fulfilled). Some iterationsare shown in Figure 3. We observed that a voltage step U < 30mV leads to convergentsolutions also for large applied voltages.
Vbarr
V(7)self, 0mV
V(0)self, 25mV
V(1)self, 25mV
V(7)self, 25mV
V(0)self, 50mV
V(1)self, 50mV
V(9)self, 50mV
−400
−200
0
200
400
potentialen
ergyin
meV
0 50 85 135position in nm
−75
−50
−25
0
25
50
75
Figure 3: Some iterations computed according to (11).
In order to compare the original Schrodinger-Poisson model with the model using ap-proximation (10), we computed the current-voltage characteristics shown in Figure 4. Here,the (conduction) current density
Jcond =e~
m
∫
R
g(k)Im
(φ∗
k
dφk
dx
)dk (12)
is approximated by a simple quadrature formula using symmetric finite differences to com-pute dφk/dx. Figure 4 shows that the results differ considerably, i.e., the choice (10) leadsto different results than those computed from the original model. Therefore, we employthe original potential energy in the transient simulations in the next subsection.
10
0
0.1
0.2
0.3
curren
tden
sity
in109A/m
2
0 0.1 0.2 0.3 0.4 0.5 0.6applied voltage in V
Figure 4: Current-voltage characteristics. The solid line corresponds to our solution ofthe original stationary Schrodinger-Poisson system. The dashed line is obtained with themodified model using approximation (10).
3. Transient simulations
In this section, we detail the numerical discretization of the transient Schrodinger equa-tions
i~∂ψk
∂t= − ~
2
2m
∂2ψk
∂x2+ V (·, t)ψk, ψk(·, 0) = φk, x ∈ [0, L], t > 0, k ∈ K, (13)
with discrete transparent boundary conditions, where K is defined in (8). To simplify thepresentation, we skip in this section the index k.
The transient Schrodinger equation (13) is discretized by the commonly used Crank-Nicolson scheme:
ψ(n+1)j−1 +
(iR− 2 + wV
(n+1/2)j
)ψ
(n+1)j + ψ
(n+1)j+1
= −ψ(n)j−1 +
(iR + 2− wV (n+1/2)
j
)ψ
(n)j − ψ
(n)j+1,
(14)
where ψ(n)j approximates ψ(xj , tn) with xj = jx and tn = nt (j ∈ Z, n ∈ N0), V
(n+1/2)j
approximates V (jx, (n+1/2)t), and R = 4m(x)2/(~t), w = −2m(x)2/~2. Underthe assumptions that the initial wave function is compactly supported in (0, L) and thatthe applied voltage vanishes, V (x, t) = 0 for x ≤ 0 and x ≥ L, t ≥ 0, it is well known (see,e.g., [4, 8]) that transparent boundary conditions for the Schrodinger equation (13) read
11
as
∂ψ
∂x(0, t) =
√2m
π~e−iπ/4 d
dt
∫ t
0
ψ(0, τ)√t− τ dτ, (15a)
∂ψ
∂x(L, t) = −
√2m
π~e−iπ/4 d
dt
∫ t
0
ψ(L, τ)√t− τ dτ. (15b)
The (homogeneous) discrete transparent boundary conditions, based on the above Crank-Nicolson scheme, are given as follows (see [3] for the derivation):
ψ(n+1)1 − s(0)ψ(n+1)
0 =
n∑
ℓ=1
s(n+1−ℓ)ψ(ℓ)0 − ψ
(n)1 , n ≥ 0, (16a)
ψ(n+1)J−1 − s(0)ψ
(n+1)J =
n∑
ℓ=1
s(n+1−ℓ)ψ(ℓ)J − ψ
(n)J−1, n ≥ 0, (16b)
with the convolution coefficients
s(n) =
(1− iR
2
)δn,0 +
(1 + i
R
2
)δn,1 + αe−inϕPn(µ)− Pn−2(µ)
2n− 1(17)
and the abbreviations
ϕ = arctan4
R, µ =
R√R2 + 16
, α =i
24√R2(R2 + 16)eiϕ/2.
Here, Pn denotes the nth-degree Legendre polynomial (P−1 = P−2 = 0), and δn,j is theKronecker symbol. In practice, the coefficients defined in (17) are computed with an effi-cient three-term recursion, relying on the three-term recursion of the Legendre polynomials[16]. The Crank-Nicolson scheme along with these discrete boundary conditions yields anunconditionally stable discretization which is perfectly reflection-free [3, 4].
Next, let the initial wave function be a solution to the stationary Schrodinger equationwith energy E and let the exterior potential at the right contact be given by a time-dependent function, V (x, t) = −eU(t) for x ≥ L, t ≥ 0. This leads to nonhomogeneoustransparent boundary conditions [1]. We describe our strategy to discretize these boundaryconditions. Our approach is motivated by that presented in [10, Appendix B], but wesuggest, similarly as in [4], a discretization of the gauge change which is compatible withthe underlying finite-difference scheme. Additionally, our approach requires only a singleset of convolution coefficients instead of two.
First, we derive a nonhomogeneous discrete transparent boundary condition at xj =L. To this end, we consider the difference between the unknown wave function ψ andthe time evolution of the scattering state, φ(x) exp(iEt/~), in the right lead [L,∞). Weemploy a gauge change to get rid of the time-dependent potential VL(t) = −eU(t). As aconsequence, the function ψ(x, t) exp(i
∫ t
0VL(s)ds/~) solves the free transient Schrodinger
equation in [L,∞). Using a similar gauge change, a straightforward computation shows
12
that φ(x) exp(−i(E − VL(0))t/~) solves the free Schrodinger equation in [L,∞) as well.Hence,
ϕ(x, t) := ψ(x, t) exp
(i
~
∫ t
0
VL(s)ds
)− φ(x) exp
(− i~(E − VL(0))t
)(18)
for x ∈ [L,∞) solves the free transient Schrodinger equation. Furthermore, ϕ(x, 0) = 0for all x ∈ [L,∞). Therefore, we could apply (15b) to derive a nonhomogeneous trans-parent boundary condition at the right contact. Instead we replace ϕ(x, t) by some ap-
proximation ϕ(n)j and subsequently apply (16b) to derive a discrete boundary condition
compatible with the Crank-Nicolson scheme. The question is how to approximate thequantities exp(i
∫ t
0VL(s)ds/~) and exp(−i(E − VL(0))t/~). Indeed, the ad-hoc discretiza-
tion for t = nt,
exp
(i
~
∫ t
0
VL(s)ds
)≈ exp
(i
~
n−1∑
ℓ=0
V(ℓ+1/2)L t
),
exp
(− i~(E − VL(0))t
)= exp
(− i~(E − V (0)
L )nt),
(19)
where V(ℓ)L = VL(ℓt), is not derived from the underlying finite-difference discretization,
causing unphysical numerical reflections at the boundary. In principle, these reflectionscan be made arbitrarily small for t → 0. However, for practical time step sizes, thecalculation of the current density would be still distorted. Our (new) idea is to applya Crank-Nicolson discretization to a differential equation satisfied by exp(i
∫ t
0VL(s)ds/~).
Indeed, this expression solves
dε
dt(t) =
i
~VL(t)ε(t), ε(0) = 1.
The Crank-Nicolson discretization of this ordinary differential equation reads as
ε(n+1) = ε(n) +t i2~V
(n+1/2)L (ε(n+1) + ε(n)), ε(0) = 1.
This recursion relation can be solved explicitly yielding
ε(n) = exp
(2i
n−1∑
ℓ=0
arctan
(t2~V
(ℓ+1/2)L
)), n ∈ N0.
A Taylor series expansion
2i arctan
(t2~V
(ℓ+1/2)L
)=i
~V
(ℓ+1/2)L t+O((t)3)
reveals that in the limit t → 0, the ad-hoc discretization in (19) coincides with thediscrete gauge change which is derived from the Crank-Nicolson time-integration method.
13
Analogously, exp(−i(E − VL(0))t/~) needs to be replaced by
γ(n)J := exp
(2i
n−1∑
ℓ=0
arctan
(−t2~E
))exp
(2i
n−1∑
ℓ=0
arctan
(t2~V
(0)L
))
= exp
[2in
(arctan
(t2~V
(0)L
)− arctan
(t2~E
))], n ∈ N0.
Thus, the discrete analogon of ϕ in definition (18) is given by
ϕ(n)j = ψ
(n)j ε(n) − φjγ
(n)J , j ∈ 0, . . . , J, n ∈ N0.
Replacing ψ(n)j by ϕ
(n)j in (16b), we obtain the desired nonhomogeneous discrete transparent
boundary condition at xJ = L:
ψ(n+1)J−1 ǫ(n+1) − s(0)ψ(n+1)
J ǫ(n+1) = −ǫ(n)ψ(n)J−1 +
n∑
ℓ=1
s(n+1−ℓ)(ψ
(ℓ)J ǫ(ℓ) − φJγ
(ℓ)J
)
− s(0)φJγ(n+1)J + φJ−1
(γ(n+1)J + γ
(n)J
).
(20)
At the left contact x0 = 0, a nonhomogeneous boundary condition can be derived in asimilar way. Since the potential energy in the left lead is assumed to vanish, the term ε(n)
is not needed, and the boundary condition is given by
ψ(n+1)1 − s(0)ψ(n+1)
0 = −ψ(n)1 +
n∑
ℓ=1
s(n+1−ℓ)(ψ
(ℓ)0 − φ0γ
(ℓ)0
)
− s(0)φ0γ(n+1)0 + φ1
(γ(n+1)0 + γ
(n)0
),
(21)
where
γ(n)0 := exp
(−2in arctan
(t2~E
)), n ∈ N0.
We summarize: The Crank-Nicolson scheme (14) with the nonhomogeneous discretetransparent boundary conditions (20) and (21) reads as
Bψ(n+1) = Cψ(n) + d(n), (22)
where ψ(n) = (ψ(n)0 , . . . , ψ
(n)J )⊤, d = (d
(n)0 , 0, . . . , 0, d
(n)J )⊤. Furthermore, B is a tridiagonal
matrix with main diagonal (−s(0), iR− 2+wV(n+1/2)1 , . . . , iR− 2+wV
(n+1/2)J−1 ,−s(0)ε(n+1)),
upper diagonal (1, . . . , 1), and lower diagonal (1, . . . , 1, ε(n+1)); C is a tridiagonal matrix
with main diagonal (0, iR + 2 − wV(n+1/2)1 , . . . , iR + 2 − wV
3.2. Fast evaluation of the discrete convolution terms
In the subsequent simulations, scheme (22) has to be solved in each time step and forevery wave function ψ = ψk, k ∈ K. We recall that the kernel coefficients s(n) need tobe calculated only once as they do not depend on the wave number k. Let N denote thenumber of time steps. For each k ∈ K, we require order O(N) storage units and O(N2) workunits to compute the discrete convolutions in (23) and (24). For this reason, simulationswith several ten thousands of time steps are not feasible. To overcome this problem, onemay truncate the convolutions at some index, since the decay rate of the convolutioncoefficients is of order O(n−3/2) [16, Section 3.3]. The drawback of this approach is thatstill more than thousand convolution terms are necessary to avoid unphysical reflectionsat the boundaries.
The problem has been overcome in [6] by approximating the original convolution coef-ficients s(n) and calculating the approximated convolutions by recursion. More precisely,approximate s(n) by
s(n) :=
s(n), n < ν,∑Λ
ℓ=1 blq−nl , n ≥ ν,
such that
C(n)(u) :=n−ν∑
ℓ=1
s(n−ℓ)u(ℓ) ≈n−ν∑
ℓ=1
s(n−ℓ)u(ℓ) (25)
can be evaluated by a recurrence formula which reduces the numerical effort drastically.As in [6], we set ν = 2 to exclude s(0) and s(1) from the approximation. In fact, s(0) doesnot appear in the original convolutions, whereas s(1) is excluded to increase the accuracy.
Let Λ ∈ N. The set b0, q0 . . . , bΛ, qΛ is computed as follows. First, define the formalpower series
The first (at least 2Λ) coefficients are required to calculate the [Λ−1|Λ]-Pade approximation
of h, h(x) := PΛ−1(x)/QΛ(x), where PΛ−1 and QΛ are polynomials of degree Λ − 1 and
Λ, respectively. If this approximation exists, we can compute its Taylor series h(x) =s(ν) + s(ν+1)x+ · · · , and by definition of the Pade approximation, it holds that
s(n) = s(n) for all n ∈ ν, ν + 1, . . . , ν + 2Λ− 1.
It can be shown that, if QΛ has Λ simple roots qℓ with |qℓ| > 1 for all ℓ ∈ 1, . . . ,Λ, theapproximated coefficients are given by
s(n) =
Λ∑
ℓ=1
bℓq−nl , bℓ := −
PΛ−1(qℓ)
QΛ(qℓ)qν−1ℓ 6= 0, n ≥ ν, ℓ ∈ 1, . . . ,Λ. (26)
Summarizing, one first computes the exact coefficients s(0), . . . , s(ν+2Λ−1) followed bythe [Λ − 1|Λ]-Pade approximation. Then one determines the roots of QΛ, yielding the
15
numbers q1, . . . , qΛ. Finally, one evaluates (26) to find the coefficients b0, . . . , bΛ. We stressthe fact that these calculations have to be performed with high precision (2Λ− 1 mantissalength) since otherwise the Pade approximation may fail (see [6]). We employ the Pythonlibrary mpmath for arbitrary-precision floating-point arithmetics [23]. As an alternative,one may use the Maple script from [6, Appendix].
A particular feature of this approximation is that it can be calculated by recursion.More precisely, for n ≥ ν + 1, the function C(n)(u) in (25) can be written as
C(n)(u) =Λ∑
ℓ=1
C(n)ℓ (u),
withC
(n)ℓ (u) = q−1
ℓ C(n−1)ℓ (u) + bℓq
−2ℓ u(n−2), n ≥ ν + 1, C
(ν)ℓ (u) = 0.
Hence, the discrete convolutions in (23) and (24) are approximated for n ≥ ν = 2 by
n∑
ℓ=1
s(n+1−ℓ)u(ℓ) ≈ C(n+1) (u) + s(1)u(n), (27)
whereas the exact expressions are used for n = 0 and n = 1. As a result, the storage forthe implementation of the discrete transparent boundary conditions reduces from O(N) toO(Λ). Even more importantly, the work is of order O(ΛN) instead of O(N2).
Obviously, the quality of the approximation depends on Λ. By construction, we haves(n) = s(n) for all n ∈ 0, . . . , 2Λ + ν − 1 but s(n) approximates s(n) very well even if n ismuch larger [6]. We illustrate in Section 4.3 that the convergence of the complete transientalgorithm with respect to Λ is exponential.
3.3. The complete transient algorithm
In the previous sections, we have explained the approximation of the transient Schro-dinger equation with discrete transparent boundary conditions for given potential energyV = Vbarr+Vself . Here, we make explicit the coupling procedure with the Poisson equationfor the selfconsistent potential
According to the Crank-Nicolson scheme, a natural approach would be to employ a two-step predictor-corrector scheme. More precisely, let ψ(n)
k k∈K → ψ(∗)k k∈K be propagated
for one time step using V(n)self to obtain V
(∗)self . Then one uses V
(n+1/2)self := 1
2(V
(n)self + V
(∗)self ) to
propagate ψ(n)k k∈K → ψ
(n+1)k k∈K again. This procedure doubles the numerical effort and
16
is computationally too costly. As an alternative, the scheme V(n+1/2)self := 2V
(n)self − V
(n−1/2)self
can be employed (as in [31]). We found in our simulations that the most simple approach,
V(n+1/2)self := V
(n)self , gives essentially the same results as the above schemes. The reason is
that the electron density evolves very slowly compared to the small time step size which isneeded to resolve the fast oscillations of the wave functions. Hence, the variations of Vselfare small. Similarly, the right boundary condition of the Poisson equation can be replacedby Vself(L) = −eU(nt) if the applied voltage varies slowly. This is used in the circuitsimulations of Section 5.
The complete transient algorithm is presented in Figure 5.
input: V(n=0)self ← Vself (φkk∈K) , ψ(n=0)
k k∈K ← φkk∈K
V (n) ← Vbarr + V(n)self
ψ(n+1)k k∈K ← ψ(n)
k k∈K
n[Vself ]←k∑
k∈K g(k)|ψ(n+1)k |2
− d2
dx2V(n+1)self = e2
ǫ(n[Vself ]− nD) , V
(n+1)self (0) = 0 , V
(n+1)self (L) = −eU (n+1)
n · dt < tfinal
V(n)self ← V
(n+1)self
stop
yes
no
Figure 5: Flow chart of the transient scheme.
3.4. Discretization parameters
We choose K = 3000 for the number of wave functions as in the stationary simulationsand t = 1 fs (fs = femtosecond) for the time step size. With the maximal kinetic energyof injected electrons ~ωM = ~
2k2M/(2m), where kM is the maximal wave number, the periodis computed according to τM = 2π/ωM . Thus, the fastest wave oscillation is resolved byτM/t ≈ 18.5 time steps. The space grid size is chosen to be x = 0.1 nm. Consequently,the smallest wave length λM = 2π/kM ≈ 10 nm is resolved by approximately 100 spatial
17
grid points. Furthermore, we take Λ = 70 for the approximation parameter of the discreteconvolution terms. This choice results from a numerical convergence study presented inSection 4.3.
It is important to note that the wave functions which are propagated using the fastevaluation of the approximated discrete convolution terms (27) practically coincide with thewave functions which are propagated using the exact convolutions (23)–(24) (see Section4.3). Employing the exact convolutions, however, is equivalent to solving the Crank-Nicolson finite difference equations of the whole space problem. Considering that theelectron density evolves smoothly in space and time, it is clear that the error of the completetransient algorithm (see Section 3.3) is determined by the Crank-Nicolson finite differencescheme. A global error estimate, together with a meshing strategy depending on a possiblyscaled Planck constant ~ is given in [7]. The calculations in this article are performed inSI units without any scaling.
3.5. Details of the implementation
The final solver is implemented in the C++ programming language using the matrixlibrary Eigen [22] for concise and efficient computations. As we are interested in simulationswith a very large number of time steps N (e.g., N = 100 000), some sort of parallelizationis indispensable. We employ the library pthreads to realize multiple threads on multi-core processors with shared memory. The most time consuming part in the transientalgorithm (see Section 3.3) is the propagation of the wave functions and the calculationof the electron density. Since the wave functions evolve independently of each other, thistask can be easily parallelized. At every time step, we create a certain number of threads(usually, this number equals the number of cores available). To each thread, we assign asubset of wave functions which are propagated as described above. Before the threads arejoined again, each thread computes its part of the electron density. All these parts providethe total eletron density which is used to solve the Poisson equation in serial mode. Thesimulations presented below have been carried out on an Intel Core 2 Quad CPU Q9950with 4× 2.8GHz.
4. Numerical experiments
We present three numerical examples. The first example shows the importance toprovide a complete compatible discretization of the open Schrodinger-Poisson system. Thesecond numerical test shows the time-dependent behavior of a resonant tunneling diode,which allows us to identify three physical regions. In the third experiment, we investigatethe convergence of our solver with respect to the parameter Λ which appears in the contextof the fast evaluation of the discrete convolution terms.
4.1. First experiment: Constant applied voltage
We compute the stationary solution to the Schrodinger-Poisson system with an appliedvoltage of U = 250mV. At this voltage, the current density achieves its first local maximum.We apply the transient algorithm of Section 3 until t = 25 fs, keeping the applied voltage
18
constant. Accordingly, the stationary solution should be preserved and the current densityJcond, defined in (12), is expected to be spatially constant.
The ad-hoc discretization (19) is employed using the time step sizes t = 1 fs, 0.5 fs,0.25 fs. We observe in Figure 6 that the current density is not constant. The reason isthat the discretization (19) is not compatible with the underlying finite-difference scheme.The distortions are reduced for very small time step sizes but this leads to computationallyexpensive algorithms. In contrast, with the discrete gauge change of Section 3.1, the currentdensity is perfectly constant even for the rather large time step t = 1 fs; see Figure 6.
dt = 1 fsdt = 0.5 fsdt = 0.25 fsdt = 1 fs
0
0.5
1
1.5
curren
tden
sity
in109A/m
2
0 50 85 135position in nm
Figure 6: Conduction current density in a resonant tunneling diode at t = 25 fs for aconstant applied voltage of U = 250mV. Discretizations using the ad-hoc discretization(19) of the analytical boundary conditions yield strongly distorted numerical solutions(broken lines). In contrast, the conduction current density computed with our solver isperfectly constant (solid line).
We mention that the transient solution is also distorted if the scattering states as initialwave functions are computed from an ad-hoc discretization of the continuous boundary con-ditions (5) and (6). For stationary computations, spurious reflections due to an inconsistentdiscretization play a minor role but they become a major issue for transient simulations.
4.2. Second experiment: Time-dependent applied voltage
For the second numerical experiment, we consider a time-dependent applied voltage.The conduction current density is no longer constant but the total current density isexpected to be conserved. We recall that the total current density Jtot = Jcond + ∂D/∂tis the sum of the conduction current density Jcond and the displacement current density∂D/∂t. Here D denotes the electric displacement field which is related to the electricfield E by D = ǫ0ǫrE. Indeed, replacing the electric field by the negative gradient of thepotential we obtain
∂D
∂t= −ǫ0ǫr
e
∂
∂t∇Vself .
19
The temporal and spatial derivatives are approximated using centered finite differences.Ampere’s circuital law ∇×H = Jtot for the magnetic field strength H yields
div Jtot = div(∇×H) = 0,
and hence, in one space dimension, Jtot is constant in space.The following simulation demonstrates that the total current density is a conserved
quantity in the discrete system as well. First, we compute the equilibrium state using anapplied voltage of U = 0V. This solution is then propagated using a raised cosine functionfor the applied voltage
U(t) =U0
2
(1− cos
2πt
T
), 0 ≤ t ≤ 1 ps,
where U0 = 0.25V and T = 2ps. At later times, t ≥ 1 ps, U(t) = U0 is kept constant.Conduction, displacement, and total current density at different times are depicted in theleft column of Figure 7. As can be seen, the total current density is perfectly conservedat all considered times. The change of the charge density ∂ρ/∂t is illustrated in the rightcolumn of Figure 7. In our model, ρ is given by ρ = e (nD − n).
The time-dependence of the total current density in response to the applied voltage isshown in Fig. 8. We can identify three different regions in the temporal behaviour, eachof which is governed by a different physical mechanism.
Region I: Capacitive behavior. When the applied voltage increases during the first picosec-ond, the resonant tunneling diode behaves mainly like a parallel plate capacitor. This canbe clearly seen in the top left panel of Figure 7. In the region of the double barrier, thedisplacement current gives the dominant contribution to the total current, whereas theconduction current is small. The top right panel of Figure 7 shows a build-up of negativecharge before the left barrier and of positive charge after the right barrier. The formationof opposite charges on the two sides of the double barrier results in the formation of anelectric field between the two regions of opposite charge density. This field is necessary toaccomodate the externally applied voltage. Figure 8 shows that the current closely followsthe time derivative of the applied voltage:
Jcond ≈ CdU
dt=πCU0
Tsin
(2πt
T
).
This expression allows us to estimate the apparent capacitance C. The maximum currentdensity occurring at t = T/4 = 0.5 ps takes approximately the value 1.2 · 109Am−2. Wecompute C = TJ/πU0 = 3.06 · 10−3 Fm−2. Equating this value to the parallel platecapacitance, C = ε0εr/d, we find the average separation of the opposite charge densitiesto be d = 33.1 nm.
Region II: Plasma oscillations. During the second picosecond, a strongly damped oscilla-tion occurs in the current density. From Figure 8, we estimate five oscillations to occurduring one picosecond, which relates to a period of about 200 fs. It is believed that these
20
0
0.5
1
1.5
curren
tden
sity
in109A/m
2
0 50 85 135position in nm
t = 0.5 ps
−0.2
−0.1
0
0.1
0.2
∂ρ/∂tin
1018C/sm
3
0 50 85 135position in nm
t = 0.5 ps
J + ∂D/∂t
J
∂D/∂t
0
0.2
0.4
curren
tden
sity
in109A/m
2
0 50 85 135position in nm
t = 1.5 ps
−0.02
−0.01
0
0.01
0.02
∂ρ/∂tin
1018C/sm
3
0 50 85 135position in nm
t = 1.5 ps
0
0.2
0.4
curren
tden
sity
in109A/m
2
0 50 85 135position in nm
t = 3.0 ps
−0.02
−0.01
0
0.01
0.02
∂ρ/∂tin
1018C/sm
3
0 50 85 135position in nm
t = 3.0 ps
Figure 7: Left column: Total current density Jtot = Jcond + ∂D/∂t, conduction currentdensity J := Jcond, and displacement current density ∂D/∂t versus position at differenttimes. Right column: Temporal variation ∂ρ/∂t of the charge density versus position.
21
Jtot
U
0
0.5
1
1.5
curren
tden
sity
in109A/m
2
0 1 2 3 4 5 6 7 8time in ps
0
0.1
0.2
0.3
0.4
voltagein
V
Jtot
U0
0.05
0.1
0.15
curren
tden
sity
in109A/m
2
0 1 2 3 4 5 6 7 8time in ps
0
0.1
0.2
0.3
0.4
voltagein
V
Figure 8: Applied voltage and total current density versus time in different scalings.
are plasma oscillations which were excited by the rapidly changing applied voltage U . Assoon as the transient phase of U(t) is over and U(t) is kept constant at U0 for t ≥ 1 ps, theexcitation vanishes and the oscillations fade out quickly. As a rough estimate we calculatethe plasma frequency ωp for a classical electron system of uniform density:
ω2p =
ne2
mε0.
Note that in the resonant tunneling diode the density is neither uniform nor is it governedby the classical equations of motion. Nevertheless, we may use this expression to estimatethe order of magnitude of the time constant associated with this effect. Since plasmaoscillations usually occur in the high-density regions of a device, we set n = n1
D = 1024m−3
and obtain τp = 2π/ωp = 111.4 fs. This value is of the same order as the 200 ps estimatedabove, which is a strong indication that the physical effect observed here is a plasmaoscillation.
Region III: Charging of the quantum well. For t > 2 ps, an exponential increase in thecurrent can be clearly observed in Figure 8. Below 2 ps we see a superposition of both theexponential current increase and the plasma oscillations. The origin of this effect can be
22
|N −N∞|
Ce−t/τ
Ua2,a5
10−4
10−3
10−2
10−1
100
|N−
N∞|in
1015/m
2
0 1 2 3 4 5 6 7 8time in ps
0
0.1
0.2
0.3
0.4
voltagein
V
Figure 9: Number of electrons in the quantum well versus time. In Region III (t ≥ 2 ps)this number clearly follows an exponential law. Ua2,a5 denotes the temporal variation ofthe voltage between x = a2 and x = a5.
understood from the right panels of Figure 7. Negative charge builds up in the quantumwell. This charge results from electrons tunneling through the left barrier into the quantumwell. In this context, we note that the temporal variation of the voltage between the leftand right end points a2 and a5 of the double-barrier structure, respectively, follows closelythe variation of the applied voltage U and hence, it is practically constant for t > 1 ps(see Figure 9). The rate |∂ρ/∂t| decreases with time as can be seen by the snapshots att = 1.5 ps and t = 3ps. We calculate the number of electrons residing in the quantum well:
N(t) :=
∫ a5
a2
n(x, t) dx.
Since the charging process is expected to show an exponential time dependence, we assumethe following exponential law for N(t) and extract the free parameters τ and N∞:
N(t) = N∞ + (N(t1)−N∞) e−(t−t1)/τ .
In Figure 9, the difference |N(t)−N∞| is plotted, which decays to zero with an extractedtime constant of τ = 1.25 ps.
This time scale is related to the life time of a quasi-bound state. At U = 0.25V, thecurrent-voltage characteristic has its first maximum, which means that the first resonantstate in the quantum well is carrying the current. The life time of this resonant state canbe extracted from the width of the resonance peak in the transmission coefficient. Figure10 depicts the transmission coefficient of the double-barrier structure at U = 0V andU = 0.25V. The transmission coefficient is defined as the ratio between the transmittedand the incident probability current density jtrans and jinc. In terms of the amplitude andthe wavenumber of the transmitted and the incident wave, it reads:
|jtrans||jinc|
=|Atrans|2ktrans|Ainc|2kinc
.
23
10−5
10−4
10−3
10−2
10−1
100
101
transm
issioncoeffi
cien
t
0.2 0.4 0.6 0.8 1energy in eV
0.2
0.4
0.6
0.8
1
transm
issioncoeffi
cien
t
0.033 0.0335 0.034 0.0345 0.035energy in eV
E1 E2
Figure 10: Transmission coefficient of the double-barrier structure at U = 0.25V in differ-ent scalings.
Extracting E, the half width at half maximum of the first transmission peak, the lifetime of the resonant state can be estimated as follows [25]:
τ =~
2E.
At U = 0.25V we find 2E = 5.31 · 10−4 eV and thus τ = 1.24 ps. This value is very closeto the time constant of τ = 1.25 ps extracted from the exponential charge increase in thequantum well, which is the cause for the observed exponential current increase.
4.3. Third experiment: Convergence in Λ
Finally, we study the convergence of the complete transient algorithm detailed in Sec-tion 3.3 with respect to the parameter Λ which appears in the context of the fast evaluationof the discrete convolution terms. For this purpose, we repeat the last experiment withdifferent values of Λ. We compare the results with those obtained from the algorithm whichuses the discrete transparent boundary conditions with the exact convolutions (23)–(24).Since the computation of the reference solution is extremely expensive, we restrict theexperiment to the final time t = 1.5 ps. The conduction current densities at t = 1.5 ps fortwo different values of Λ and for the reference solution are depicted in Figure 11 (left). Therelative error in the ℓ2-norm for increasing values of Λ is shown in Figure 11 (right). Weobserve that the relative error decreases exponentially fast. Thus, using a relatively smallvalue of Λ practically yields the same results (at dramatically reduced numerical costs) asif the discrete transparent boundary conditions with the exact convolutions were used.
5. Circuit simulations
In this section, we simulate a high-frequency oscillator consisting of a voltage sourceUe, a resistor with resistance R, an inductor with inductance L, a capacitor with capacity
24
Λ = 10Λ = 20ref
0
0.05
0.1
0.15
0.2
curren
tden
sity
in109A/m
2
0 50 85 135position in nm
10−10
10−8
10−6
10−4
10−2
100
rel.
error
0 20 40 60 80 100 120Λ
Figure 11: Conduction current density at t = 1.5 ps (left) and relative ℓ2-error for increasingΛ.
C, and a resonant tunneling diode RTD; see Figure 12. Each element of the circuit yieldsone current-voltage relationship,
UR = RIR, UL = LIL, IC = CUC , IRTD = f(URTD). (28)
The last expression is to be understood as follows. Given the applied voltage URTD atthe tunneling diode, the current IRTD(t) = AJtot(t) is computed from the solution of thetime-dependent Schrodinger-Poisson system. Here, A = 10−11m2 is the cross sectionalarea of the diode and Jtot is the total current density. In the simulations we use R = 5Ω,L = 50 pH, and C = 10 fF.
L
R
RTD CUe
Figure 12: High-frequency oscillator containing the resonant tunneling diode RTD.
According to the Kirchhoff circuit laws, we have
Ue = UR + URTD + UL, URTD = UC , IL = IR, IL = IRTD + IC . (29)
25
Combining (28) and (29), we find that
CURTD = CUC = IC = IL − IRTD,
LIL = UL = Ue − UR − URTD = Ue −RIR − URTD = Ue −RIL − URTD.
Consequently, we obtain a system of two coupled ordinary differential equations,
d
dt
(URTD
IL
)=
(0 1
C
− 1L−R
L
)(URTD
IL
)+
(− 1
CIRTD
1LUe(t)
). (30)
The time-step size t is very small compared to the time scale of the variation of thepotential energy and the variation of the current flowing through the diode. Hence, usingthe same time step for the time integration of (30), we can resort to an explicit time-stepping method. We choose the simplest one, the explicit Euler method. Alternatively,one may employ an implicit method, but we observed that both methods yield essentiallythe same results.
First circuit simulation. In the first simulation, the RTD solver is initialized with the steadystate corresponding to URTD(t) = 0 for all t ≤ 0. The external voltage Ue is assumed tobe zero for t ≤ 0, and the initial conditions for (30) are URTD(0) = 0 and IL(0) = 0.For t ∈ [10, 20] ps, the external voltage is increased smoothly to 0.275V and then keptconstant (see Figure 13). This value is between the voltages where the stationary currentdensity reaches its local maximum and minimum (see Figure 4). The time evolution of thevoltage and the current at the RTD are depicted in Figure 13. It is clearly visible thatthe system starts to oscillate. Furthermore, the potential energy, electron density, currentdensities, and the temporal variation of the total charge ∂ρ/∂t are shown in Figure 14 forfour different times from the interval [t1 = 77.7, t2 = 87.2] ps, which covers exactly oneoscillation. Around 2 ps after the beginning of the period, the electron density within thequantum well in [65, 70] nm becomes minimal (first row). After some time, we observe abuild-up of negative charge in the quantum well with ∂ρ/∂t < 0 (second row). At aboutt = 84.6 ps the electron density reaches its maximum value (third row). Subsequently, theelectrons leave the quantum well again and ∂ρ/∂t > 0 in [65, 70] nm (fourth row). Thefrequency of the oscillations is approximately 105GHz which corresponds qualitativelyto frequencies observed in standard double-barrier tunneling diodes [14]. The temporalevolution of the physical quantities in the circuit is animated in the video available athttp://www.asc.tuwien.ac.at/~mennemann/projects.html.
Second circuit simulation. In this experiment, the external voltage Ue is kept fixed for alltimes. At times t ≤ 0, the circuit contains the voltage source, resistor, and RTD only. Weinitialize the transient Schrodinger-Poisson solver with the steady state corresponding toURTD(t) = 0.275V for all t ≤ 0. To compensate for the voltage drop at the resistor, theexternal voltage is set to
Ue(t) = RIRTD(t) + URTD(t), t ≤ 0.
26
Ue
URTD
IRTD
−0.1
0
0.1
0.2
0.3
0.4
voltagein
V
0 25 50 75 100time in ps
t1 t2
−0.4
0
0.4
0.8
curren
tin
10−2A
Figure 13: First circuit simulation: Voltage URTD and current IRTD through the resonanttunneling diode versus time.
At time t = 0, the capacitor and inductor are added to the circuit. In order to avoiddiscontinuities in the voltages, we charge the capacitor with the same voltage wich isapplied at the RTD before the switching takes place, UC(t) = URTD(t) for t ≤ 0. Forsimilar reasons, we set the current flowing through the inductor to the current flowingthrough the RTD, IL(t) = IRTD(t) for t ≤ 0. This configuration corresponds to theequilibrium state. Therefore, one would expect that the system remains in its initial statefor all time. However, the equilibrium is unstable and a small perturbation will drive thesystem out of equilibrium. In fact, numerical inaccuracies suffice to start the oscillator.However, we accelerate the transient phase by perturbing IL(t) by the value 5 · 10−6A fort ≤ 0. The numerical result is presented in Figure 15. The simulation took less than 4hours computing time on an Intel Core 2 Quad Core Q9950 with 4× 2.8 GHz.
Acknowledgements
The first two authors acknowledge partial support from the Austrian Science Fund(FWF), grants P20214, P22108, and I395, and the Austrian-French Project of the AustrianExchange Service (OAD).
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n
V0
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28
URTD
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0
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tin
10−2A
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